We study the truncation functors and show the existence of projective cover of each irreducible module in parabolic BGG category $\mathcal O$ over infinite rank Lie algebra of types $\mathfrak{ a,b,c,d}$. Moreover, $\mathcal O$ is a Koszul category. As a consequence, the corresponding parabolic BGG category $\overline{ \mathcal O}$ over infinite rank Lie superalgebra of types $\mathfrak{ a,b,c,d}$ through the super duality is also a Koszul category.
Categories: math.RT